If `iz^(3) + z^(2) -z+i=0`, then `|z|` is equal to

Options

A. 1
B. i
C. `-1`
D. `-i`

Answer

To solve the equation \( iz^3 + z^2 - z + i = 0 \) for \( |z| \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ iz^3 + z^2 - z + i = 0 \]
We can rearrange it to isolate the terms involving \( z \):
\[ iz^3 + z^2 - z = -i \]

### Step 2: Factor Out Common Terms
Next, we can factor the left-hand side:
\[ z^2(iz + 1) - z = -i \]
This can be rewritten as:
\[ z^2(iz + 1) = z - i \]

### Step 3: Solving the Factorized Equation
We can set up two equations based on the factorization:
1. \( z^2 = -i \)
2. \( iz + 1 = 0 \)

### Step 4: Solve for \( z \) from \( z^2 = -i \)
To find \( z \), we take the square root of both sides:
\[ z = \pm \sqrt{-i} \]
To simplify \( \sqrt{-i} \), we can express \( -i \) in polar form. The modulus of \( -i \) is 1, and its argument is \( -\frac{\pi}{2} \):
\[ -i = e^{-i\frac{\pi}{2}} \]
Thus,
\[ \sqrt{-i} = e^{-i\frac{\pi}{4}} = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \]
So, we have:
\[ z = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \quad \text{or} \quad z = -\left(\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}\right) \]

### Step 5: Calculate the Modulus of \( z \)
Now, we calculate the modulus \( |z| \):
\[ |z| = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \]

### Step 6: Conclusion
Thus, the modulus \( |z| \) is:
\[ |z| = 1 \]